Determine Whether Each Function Is Even, Odd, Or Neither.1. F ( X ) = X 2 − 9 F(x) = \sqrt{x^2} - 9 F ( X ) = X 2 ​ − 9 - $\square$2. G ( X ) = ∣ X − 3 ∣ G(x) = |x-3| G ( X ) = ∣ X − 3∣ - $\square$3. F ( X ) = X X 2 − 1 F(x) = \frac{x}{x^2 - 1} F ( X ) = X 2 − 1 X ​ - $\square$4. $g(x) = X +

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In mathematics, functions are classified as even, odd, or neither based on their behavior when input values are negated. This classification is crucial in various mathematical operations and applications. In this article, we will determine whether each of the given functions is even, odd, or neither.

What are Even, Odd, and Neither Functions?

Before we dive into the examples, let's understand the definitions of even, odd, and neither functions.

  • Even Functions: A function f(x) is even if f(-x) = f(x) for all x in the domain of the function. This means that the function's graph is symmetric with respect to the y-axis.
  • Odd Functions: A function f(x) is odd if f(-x) = -f(x) for all x in the domain of the function. This means that the function's graph is symmetric with respect to the origin.
  • Neither Functions: A function that does not satisfy the conditions for even or odd functions is classified as neither.

Example 1: f(x)=x2−9f(x) = \sqrt{x^2} - 9

Let's determine whether the function f(x) = \sqrt{x^2} - 9 is even, odd, or neither.

To check if the function is even, we substitute -x for x and simplify the expression:

f(-x) = \sqrt{(-x)^2} - 9 f(-x) = \sqrt{x^2} - 9

Since f(-x) = f(x), the function f(x) = \sqrt{x^2} - 9 is even.

Example 2: g(x)=∣x−3∣g(x) = |x-3|

Next, let's determine whether the function g(x) = |x-3| is even, odd, or neither.

To check if the function is even, we substitute -x for x and simplify the expression:

g(-x) = |(-x)-3| g(-x) = |-x-3| g(-x) = |x+3|

Since g(-x) ≠ g(x), the function g(x) = |x-3| is neither even nor odd.

Example 3: f(x)=xx2−1f(x) = \frac{x}{x^2 - 1}

Let's determine whether the function f(x) = \frac{x}{x^2 - 1} is even, odd, or neither.

To check if the function is even, we substitute -x for x and simplify the expression:

f(-x) = \frac{-x}{(-x)^2 - 1} f(-x) = \frac{-x}{x^2 - 1} f(-x) = -\frac{x}{x^2 - 1}

Since f(-x) = -f(x), the function f(x) = \frac{x}{x^2 - 1} is odd.

Example 4: g(x)=x+1xg(x) = x + \frac{1}{x}

Finally, let's determine whether the function g(x) = x + \frac{1}{x} is even, odd, or neither.

To check if the function is even, we substitute -x for x and simplify the expression:

g(-x) = -x + \frac{1}{-x} g(-x) = -x - \frac{1}{x}

Since g(-x) ≠ g(x), the function g(x) = x + \frac{1}{x} is neither even nor odd.

Conclusion

In conclusion, we have determined the even, odd, or neither classification of each of the given functions. The functions f(x) = \sqrt{x^2} - 9 and f(x) = \frac{x}{x^2 - 1} are even and odd, respectively. The functions g(x) = |x-3| and g(x) = x + \frac{1}{x} are neither even nor odd.

Key Takeaways

  • Even functions satisfy the condition f(-x) = f(x) for all x in the domain.
  • Odd functions satisfy the condition f(-x) = -f(x) for all x in the domain.
  • Functions that do not satisfy the conditions for even or odd functions are classified as neither.

Real-World Applications

Understanding even, odd, and neither functions is crucial in various mathematical operations and applications, such as:

  • Trigonometry: Even and odd functions are used to represent periodic functions, such as sine and cosine.
  • Calculus: Even and odd functions are used to determine the symmetry of functions and their derivatives.
  • Signal Processing: Even and odd functions are used to represent signals and their properties.

By understanding the classification of functions as even, odd, or neither, we can better analyze and apply mathematical concepts in various fields.